Experience First

Students start unit 4 by recalling ideas from Geometry about right triangles. They consider the relative size of sides in a right triangle and relate this to the measure of the angle across from it. The goal of today’s lesson is that students grasp the concept that angles in a right triangle determine the ratio of sides and that these ratios have specific names, namely sine, cosine, and tangent.

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You may wish to project the lesson onto a screen so that students can see the colors of the sides if they are using black and white copies. In question 4, make sure students write the answers as fractions and decimals. Fractions emphasize the comparison of sides and decimals emphasize the equivalence of the ratios. Throughout this unit we will continue to point out that a decimal can also denote a comparison of two sides and not just one singular quantity.

For question 6, students are likely to say that the sine ratio will stay the same since both the opposite side and the hypotenuse are increasing. It is not immediately evident to them that they would not change by the same amount, thus altering the ratio. Give students time to wrestle through this idea and pose questions such as “How do you know sine will stay the same? Can you give me a convincing argument?”

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Formalize Later

The use of the word “ratio” is important throughout this entire unit. It is critical that students understand that even a decimal value can represent a comparison of two sides. Some of the check your understanding questions are centered around this idea of interpreting decimals as comparisons (question 4 and 5). 

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It is also important to emphasize that knowing for example that the sine of an angle is 7/18 does not necessarily imply that the opposite side is 7 and the hypotenuse is 18, simply that 7/18 represents the ratio of sides

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In this lesson we primarily use the phrase trig ratios rather than trig functions, but this shift will happen throughout the unit especially as we look at the graphs of the trig functions in lessons 4.5 and beyond.